If I recall, you have a private pro app, so you should be able to use the Send Email action. I’ve never used it myself, so I would assume that it should work. I would just make sure that you haven’t exceeded any monthly limits. For something to try, submit your form, but instead of an on submit action, view the details for that new row and add a button with the send email action. This would be a good way to isolate is there is an issue with the send email option, or with using it on a form submit. Also, does your recipient email come from a computed column, or is it written to a basic column when the form is submitted?
Overall, no, you should not need to integrate something like Zapier in this case.